Solve for $z$, $ \dfrac{4z + 7}{3z + 6} = -\dfrac{9}{z + 2} - \dfrac{4}{5z + 10} $
First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3z + 6$ $z + 2$ and $5z + 10$ The common denominator is $15z + 30$ To get $15z + 30$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{4z + 7}{3z + 6} \times \dfrac{5}{5} = \dfrac{20z + 35}{15z + 30} $ To get $15z + 30$ in the denominator of the second term, multiply it by $\frac{15}{15}$ $ -\dfrac{9}{z + 2} \times \dfrac{15}{15} = -\dfrac{135}{15z + 30} $ To get $15z + 30$ in the denominator of the third term, multiply it by $\frac{3}{3}$ $ -\dfrac{4}{5z + 10} \times \dfrac{3}{3} = -\dfrac{12}{15z + 30} $ This give us: $ \dfrac{20z + 35}{15z + 30} = -\dfrac{135}{15z + 30} - \dfrac{12}{15z + 30} $ If we multiply both sides of the equation by $15z + 30$ , we get: $ 20z + 35 = -135 - 12$ $ 20z + 35 = -147$ $ 20z = -182 $ $ z = -\dfrac{91}{10}$